Shockwave attenuation characteristics of rapid-assembly anti-blast walls under explosive load

Abstract

This study focused on the anti-explosion performance of rapidly assembled modular blast walls under a 117.5 kg TNT explosion. The test recorded diffracted overpressure on the rear side of the wall, revealing a nonlinear spatial distribution of overpressure behind the wall. Under the combined effects of Mach reflection and wave system superposition, a secondary pressure peak occurred at a position 1.0 times the wall height behind the wall. A numerical model validated by experiments was established using LS-DYNA software to quantitatively analyze the influence weights and threshold effects of wall proportional height, proportional distance, and proportional thickness on protective effectiveness. For the first time, this study reveals the unique four-peak diffraction mechanism for this specific configuration, elucidating the local overpressure enhancement law induced by Mach reflection at 1.0 times the wall height behind the wall. By combining cranial injury criteria, the safe area behind the wall was qualitatively divided for this working condition. An empirical formula for predicting overpressure was established, comprehensively considering the coupling effects of TNT equivalent, blast wall geometric parameters, and distance behind the wall. The results show that the maximum overpressure reduction rate of this type of blast wall is 78.2%, and the average near-field protection effectiveness is approximately 1.76 times that of the far field. The research findings can offer valuable references and optimization strategies for protective structures against hundred-kilogram-level explosion impact.

Graphical Abstract

Keywords

rapid-assembly anti-blast walls diffracted overpressure mach reflection explosion protection safety zone

Highlights

• Field test with 117.5 kg TNT measured the diffracted overpressure behind the wall, showing a maximum attenuation of 78.2%;

• Numerical analysis quantified wall height as the dominant protective parameter, exhibiting a threshold effect;

• The unique four-peak diffraction mechanism of the variable-section wall was revealed and explained;

• An empirical formula for predicting peak overpressure behind the wall was established.

Introduction

Explosions pose a dominant threat to modern protective engineering, inducing catastrophic structural collapse and large-scale human casualties. The core destructive mechanism stems from high-pressure impulses and energy dissipation of blast-induced shockwaves. Consequently, quantifying the attenuation dynamics of shockwaves through air media is essential for advancing blast mitigation strategies and refining the design of protection systems.

Statistics indicate that approximately 70% of blast-related casualties in modern military operations result from intentional explosive incidents (e.g., improvised explosive devices, artillery shells), with the remaining 30% caused by unexploded ordnance, ammunition handling accidents, and other unintended explosive-related events (Blum and DeBruyne, 2020). Blast barriers serve as primary countermeasures, with major types including concrete walls, water-filled barriers, grating-type barriers, and rapid-assembly anti-blast walls. Concrete blast walls provide excellent durability and blast resistance, but they incur high costs due to material density and prolonged construction cycles. Their failures may also generate hazardous secondary fragments (Neimark et al., 2024). Water-filled barriers achieve over 70% blast wave attenuation (Yang et al., 2023); however, their practical application is severely limited by high susceptibility to environmental factors, such as freezing-induced brittleness at low temperatures, water evaporation under high-temperature exposure, UV aging of plastic containers, and fragment-induced leakage (Shirbhate and Goel, 2021). Fence type blast walls utilize wave interference for lightweight design (Jin et al., 2019), but their porous structures will accumulate damage under repeated blasts and provide inadequate fragmentation resistance (Hao et al., 2017).

Soil-filled barriers represented by rapid-assembly Anti-Blast walls demonstrate unique advantages: collapsible gabion containers filled with on-site soil allow for assembly 5 ∼ 8 times faster than that of concrete walls (Xiao et al., 2023), while reducing material cost by over 60%. Granular soil structures dissipate energy through interparticle friction, exhibiting nonlinear energy absorption capacity under blast loading (Scherbatiuk and Rattanawangcharoen, 2011), and the experimentally validated hybrid rigid-body rotation model can predict displacement-time histories (Scherbatiuk et al., 2008; Scherbatiuk and Rattanawangcharoen, 2010).

The overpressure distribution exhibits significant complexity due to the coupled effects of barrier diffraction and reflection (Wei et al., 2021). However, current research faces three critical knowledge gaps that limit the engineering application of rapid-assembly anti-blast walls: (1) Most existing studies predominantly focus on small-scale TNT detonations with charge weights less than 10 kg, while systematic full-scale test data on the shockwave attenuation mechanism and dynamic response of barriers under hundred-kilogram-scale blasts (a common scenario in terrorist attacks and accidental industrial explosions) are still scarce; (2) Despite the existence of semi-analytical models for free-field blast overpressure prediction, a systematic quantification of the coupled effects of wall height and explosive yield on the diffracted overpressure attenuation behind anti-blast walls is still lacking (Ryu et al., 2024; Yang et al., 2020); (3) Existing research fails to integrate the diffracted overpressure parameters behind barriers with quantitative human injury thresholds, resulting in non-actionable safe distance design methods for engineering practices (Denny et al., 2024).

To address these challenges, this paper is structured as follows. Field anti-blast experiments with 117.5 kg TNT charge are carried out to acquire diffraction overpressure time-history data behind rapid-assembly variable cross-section blast walls. A numerical model is built via the nonlinear dynamic software LS-DYNA and validated against measured test data. On this basis, the unique four-peak diffraction mechanism and Mach reflection effect of the blast wall are revealed. The influences of wall height, scaled distance and wall thickness on protective performance, together with their weight characteristics and threshold effects, are quantitatively analyzed. Safe regions are divided according to the craniocerebral injury criterion (Yang et al., 1996), and an empirical overpressure prediction formula considering multi-parameter coupling effect is established. The obtained results can offer quantitative references for emergency deployment of anti-terrorism protection works.

Experimental study

To quantitatively assess the dynamic protective performance of Rapid-Assembly Anti-Blast Walls under explosive loads, an anti-blast test was designed and conducted using 117.5 kg of TNT.

Experimental setup

Anti-blast wall configuration and dimensions

One set of rapid-assembly anti-blast wall specimens was employed in the experiment. The layout scheme is illustrated in Figure 1. During the experiment, the blast-facing surface of the wall was oriented directly towards the detonation centre, and the horizontal standoff distance between the front edge of the blast-facing surface and the detonation centre was set to 5 m.

Figure 1.

Configuration of the blast wall and the cylindrical TNT charge.

The anti-blast wall adopted a modular rapid-assembly structural system and is fabricated by assembling two types of standard basic units. Each basic unit employs a low-carbon steel wire welded mesh as its load-bearing skeleton. A high-strength flame-retardant geotextile is lined on the inner side of the unit, which is then densely filled with compacted backfill material to form a cubic load-bearing unit with stable stiffness. Adjacent units are connected using helical hinges to achieve rigid connection and rapid assembly. In the experiment, two types of rapid-assembly basic units, rectangular and trapezoidal, were utilized. Their specifications are as follows: the dimensions of the rectangular unit are 1060 mm (length) × 1060 mm (width) × 2210 mm (height). The cross-section of the trapezoidal unit is a right-angled trapezoid, with lengths of 1060 mm for the top base and 1800 mm for the bottom base. The unit depth (width) is 1060 mm, and the height is 2210 mm.

The overall anti-blast wall employed a double-layer assembly structure, comprising an upper and a lower layer. The upper layer was constructed by assembling two sets of rectangular units, while the lower layer was a combination of two sets of trapezoidal units and one set of rectangular units. Each set of units (a column) was formed by sequentially splicing five basic units of the same specification along the transverse direction of the wall. The overall configuration of the wall is illustrated in Figure 1. The overall longitudinal dimension (perpendicular to the detonation centre axis) of the wall is 5.3 m. The thickness is 2.12 m at the upper section and 4.66 m at the lower section, with a total height of 4.42 m.

A cylindrical bare TNT charge was employed as the explosion load source in the experiment. The configuration and installation positioning of the charge are shown in Figure 1. The charge had a total mass of 117.5 kg, an axial height of 840 mm, and a cross-sectional diameter of 315 mm. The charge was arranged in a vertical orientation. The horizontal distance between its center and the midpoint of the front edge at the base of the wall’s blast-facing surface was 5.0 m. The axis of the detonation centre coincided completely with the center normal line of the wall’s blast-facing surface, and the height of the detonation centre above the ground was set to 1.0 m. This configuration ensured that the primary propagation direction of the blast wave was perpendicular to the blast-facing surface of the anti-blast wall, fully satisfying the spatial positioning and uniformity control requirements for the blast load on the facing surface in explosion mechanics experiments.

The ground of the experimental site consisted of natural hard sandy soil, which was generally level in its natural state without artificial leveling, compaction, or hardening treatment. The surface exhibited naturally occurring slight undulations and roughness. Studies have shown that surface roughness affects the ground reflection characteristics of shock waves and modifies the formation and evolution of the Mach stem, thereby altering the peak overpressure at ground measurement points. For instance, Needham (2018) noted that rough ground increases energy dissipation during shock wave propagation, leading to ground-reflected overpressures lower than the theoretical values for smooth, rigid surfaces. Lechat et al. (2021) further confirmed through experiments and simulations that periodically rough surfaces can increase the attenuation of peak pressure for weak shock waves by 10% to 15%, and the formation height of the Mach stem increases with greater roughness. The natural sandy soil ground used in this experiment had relatively low roughness. Its influence on shock wave propagation was significantly weaker than the diffraction effect of the anti-blast wall. Therefore, the overpressure distribution behind the wall was primarily governed by the geometric parameters of the wall and the standoff distance. Furthermore, this untreated ground condition more closely resembles practical engineering scenarios such as counter-terrorism protection and industrial explosions, which enhances the engineering applicability of the experimental results.

Measurement points

Reflected overpressure was monitored at the geometric center of the blast-facing surface and at a point 1.1 m to the left of the central axis. A diffracted overpressure monitoring array was deployed on the ground behind the wall. All pressure sensors behind the wall were flush-mounted on the ground surface, with their sensitive surfaces oriented vertically upward. The layout of the measurement points is shown in Figure 2. This arrangement, utilizing an orthogonal array, enables systematic data acquisition on the characteristics of the shock wave reflection and the patterns of diffraction attenuation, facilitating subsequent analysis of the experimental data.

Figure 2.

Placement diagram of pressure sensors.

Instrumentation

A multi-channel Kistler piezoelectric sensor array was deployed to monitor the explosion pressure. The sensors on the blast-facing side were installed using a flange-embedded method, with their sensitive surfaces orthogonally aligned to the detonation axis, as shown in Figure 3. The data acquisition system was triggered by the first arrival of the shock wave at the front sensor T1, with a pre-trigger time set to 5 milliseconds. The raw overpressure signals were processed through a fourth-order Butterworth low-pass filter with a cutoff frequency of 100 kHz to eliminate high-frequency noise interference while preserving the effective characteristics of the blast shock wave.

Figure 3.

Sensors used in the test.

The ground-mounted sensors were installed flush and sealed to prevent environmental interference. Data acquisition was performed using a Tian Ao SC56671 high-speed system, with specific details listed in Table 1. The trigger signal was converted into a voltage signal via a charge amplifier, as shown in Figure 4.

Table 1.

Performance specifications of major instruments.

No.	Instrument	Specification	Detail
1	Data acquisition device	TianAo SC56671A-01	Channels: 32
			Sample rate: 10 MSa/s maximum
			Resolution: 14 bits
2	Gauge	Kistler 211B6	Range: 50 psi
			Rise Time: 2 μs
			Linearity: ±1% F.S.
3	Signal cable	STYV-2-2 (Tianjin 609 Factory)	Length: ≤80 m

Figure 4.

Data acquisition system.

Results and analysis

The experimental parameters are defined as follows: P_s and P_r denote the peak incident overpressure and the peak reflected overpressure measured in the experiment, respectively, while P_C refers to the peak incident overpressure in unobstructed free field at the corresponding scaled distance, which is calculated by the CONWEP conventional weapon explosion effect code (Kingery and Bulmash, 1984). This parameter merely acts as the benchmark reference for attenuation rate calculation. All overpressure values are expressed in megapascals (MPa). The scaled distance $\bar{R}$ (m/kg^1/3) between a measurement point and the detonation centre is defined as:

\bar{R} = \frac{R}{\sqrt[3]{W}}

(1)

where R is the distance between the measurement point and the detonation centre, m;W is the equivalent TNT mass, kg.

The blast wave attenuation rate $η$ is defined as:

η = \frac{P_{C} - P_{i}}{P_{C}} \times 100 %

(2)

where

P_{i}

is the peak transmitted diffraction overpressure captured at monitoring points behind the wall, generated by blast waves diffracting over and along the wall surface. Here, i stands for the serial number of measurement points. A higher value of

η

indicates a more significant blast wave attenuation effect at the corresponding location.

Macroscopic observations

Following detonation, the anti-blast wall did not exhibit any displacement. Affected by the high-temperature jet impact, localized charring occurred on the geotextile of the blast-facing surface, as shown in Figure 5. The internal backfill maintained its structural integrity with no collapse observed. The flame-retardant treated geotextile on the side surfaces showed only slight discoloration due to thermal radiation. Furthermore, the surface flatness was consistent with the assumption of a rigid boundary. Therefore, in numerical simulations, the wall can be simplified as a rigid boundary.

Figure 5.

Macroscopic damage under anti-nlast experiment.

Experimental data analysis

Figure 6 presents the overpressure–time history curves for each measurement point. As shown in the figure, measurement points T1 and T2 on the blast-facing surface exhibited multiple dense, saturated spikes. This occurred because the blast-facing surface was subjected not only to the strong reflection of the initial incident shock wave, but also to multiple reciprocal reflections of the shock wave between the sensor flange and the wall surface, as well as to high-frequency pressure oscillations induced by the high-velocity impact of explosion products. In addition, the lower section of the anti-blast wall employed trapezoidal units with an inclined blast-facing surface. The reflected pressure under oblique incidence differs markedly from that under normal incidence. All these signals far exceeded the measurement range of the sensors, leading to continuous waveform saturation and truncation. Therefore, data obtained from these two points are invalid and excluded from all subsequent analyses in this study.

Figure 6.

Overpressure time-history curves of measurement points.

Table 2 summarizes the peak diffracted overpressures at various measurement points behind the wall, together with the theoretical free-field overpressure values calculated using the CONWEP code at the corresponding locations. Figure 7 compares the measured diffracted overpressures behind the wall with the CONWEP-calculated free-field values.

Table 2.

Comparison of measured overpressure with free-field calculated values from CONWEP.

Gauge NO.	$\bar{R}$ (m/kg^1/3)	Experimental Value (MPa)	CONWEP(MPa)	$η$
T3	2.58	0.110	0.507	78.2%
T4	2.99	0.122	0.339	64.1%
T5	3.40	0.104	0.243	57.3%
T6	3.71	0.114	0.196	41.6%
T7	4.01	0.085	0.163	47.8%
T8	4.42	0.087	0.131	33.8%
T9	4.83	0.066	0.109	39.7%

Figure 7.

Comparison between measured diffracted overpressure and CONWEP free-field calculations.

As shown in Figure 7, the rapid-assembly anti-blast wall exhibits excellent blast wave attenuation and protection performance. The closer the measurement point is to the rear face of the wall, the more significant the overpressure reduction effect, with a measured maximum attenuation rate of 78.2%. The protective mechanism is as follows: when the strong discontinuous shock wave front generated by the explosion propagates to the anti-blast wall, its propagation path is blocked by the rigid wall. The incident shock wave undergoes intense reflection on the blast-facing surface, and the coupling of the incident and reflected waves deflects the propagation direction of the shock wave, causing the wave front to diffract around the edges of the wall towards the rear. During this process, the energy of the shock wave is significantly dissipated and redistributed, ultimately forming a large-area low-pressure shadow zone behind the wall, providing effective protection for targets located there (Hajek and Foglar, 2015).

For measurement points on the centerline behind the wall, the blast shock wave diffracts and propagates from three directions simultaneously: over the top of the wall and around both sides. The lateral diffracted waves traveling around the sides do not need to surmount the top of the wall, resulting in a shorter propagation path. Consequently, they arrive at the measurement point first, forming the lower-amplitude first-arrival wave shown in Figure 6. Subsequently, the main shock wave diffracted over the top of the wall arrives. After propagating to the ground, this wave undergoes strong reflection. The superposition of the incident and ground-reflected waves forms a Mach reflection, producing a higher peak overpressure that corresponds to the maximum main peak in Figure 6.

As the distance from the measurement point to the wall increases (from T3 to T5), the arrival time difference between the lateral diffracted wave and the top-diffracted wave gradually decreases, and the separation between the first-arrival wave and the main peak becomes more distinct. This observation is consistent with the findings of Lechat et al. (2021) regarding shock wave propagation behind barriers of finite width.

Further analysis of the overpressure data at measurement points behind the wall reveals that the peak overpressure generally decays gradually with increasing scaled distance. However, a local overpressure rebound occurs at measurement point T4. The physical mechanism is as follows: after the diffracted shock wave behind the wall is reflected by the rigid ground, the incident wave and the ground-reflected wave superimpose and couple at the wavefront, forming a Mach reflection at a location approximately one wall-height behind the wall. The superposition of this Mach reflection with the lateral diffracted wave system collectively causes the local overpressure increase at this measurement point.

Meanwhile, because the wall employed in the experiment is of finite length, the shock wave can diffract laterally around both sides of the wall. These laterally diffracted shock waves interact and superimpose with the main diffracted shock wave that passes over the top of the wall in the region behind the wall, producing a significant pressure enhancement effect. This pattern has been confirmed by comparative studies on the far-field shock wave effects of blast walls, such as those conducted by Chester et al. (2025).

In summary, the coupling and superposition effects of multiple wave systems jointly determine the spatial distribution characteristics of the overpressure behind the wall.

In addition to measurement point T4, an anomalous overpressure rebound was also observed at point T6, where the overpressure attenuation rate (41.6%) was significantly lower than that at point T5 (57.3%). Based on an analysis of the on-site experimental conditions, it is inferred that this anomaly was caused by impacting fragments splashed from the ground striking the pressure sensor during the experiment. The rebound at point T4 represents the genuine physical effect of Mach reflection behind the wall, whereas the data from point T6 is invalid due to experimental interference. Therefore, this measurement point is excluded from all subsequent analyses.

Numerical simulation

Finite element model

A multi-material Arbitrary Lagrangian-Eulerian (ALE) numerical model for air and explosive was established in the LS-DYNA software to simulate the propagation of the explosion shock wave and its fluid-structure interaction effects with the anti-blast wall. Considering the axisymmetric propagation characteristics of the explosion shock wave, a 1/4 symmetric three-dimensional model was employed to balance computational accuracy and solution efficiency.

Referring to the conclusions of the mesh sensitivity study by Shi Lei et al. (Shi et al., 2010; Wang et al., 2016), a near-field air domain mesh size not exceeding 3/80 of the charge diameter (i.e., 12.75 mm) can accurately capture the characteristics of the shock wave front, while a gradient coarsening strategy can be applied to the far-field region. Therefore, the element sizes for this numerical model were progressively increased as 1 cm, 2 cm, 4 cm, 8 cm, and 16 cm, balancing computational efficiency while ensuring accuracy (Draganic and Varevac, 2018; Souli et al., 2012). The model utilized a three-dimensional mesh mapping technique. The meshing scheme is shown in Figure 8. The air domain is defined as 26 m in both height and width, exceeding the overall dimensions of the wall. Accurate simulation of shock wave diffraction at the wall top and flanks is achieved, and the experimental boundary conditions and load cases are reproduced. The actual geometry of the 5.3 m long blast wall is fully reproduced in the numerical model, whereby high accuracy is guaranteed for the simulation of three-dimensional lateral diffraction.

Figure 8.

Schematic diagram of the 3D numerical model.

Based on the preliminary experimental results, the dynamic structural response of the anti-blast wall itself is negligible. Therefore, the wall was modeled as a rigid body in the simulation to achieve precise alignment with the experimental conditions. The TNT charge was modeled as a cylinder according to its actual dimensions from the experiment. Symmetry boundary conditions were applied to the symmetric cross-sections to reduce computational cost. Finally, the total number of elements in the model mesh was 20,027,443. Under single-node parallel computing conditions, the total computation time was approximately 17 hours.

Air was simulated using the ideal gas equation of state, while the TNT explosive was described by the JWL (Jones-Wilkins-Lee) equation of state to model the detonation process. The relevant material parameters employed are standard values commonly used in the field of blast wall research, as detailed in Tables 3 and 4.

Table 3.

Parameters related to numerical simulation of air.

ρ (kg/m³)	E (J/m³)	$V_{0}$	$γ$
1.225	1.5 × 10⁵	1.0	1.4

Symbol definitions in the table: ρ denotes air density, (kg/m³); E represents internal energy per unit volume (kJ/m³); $V_{0}$ is initial relative volume (dimensionless); $γ$ stands for specific heat ratio (dimensionless).

Table 4.

Parameters related to numerical simulation of explosives.

ρ (kg/m³)	D (m/s)	A (Pa)	B (Pa)	ω	$R_{1}$	$R_{2}$
1791	7980	5.242 × 10¹¹	7.678 × 10⁹	0.35	3.42	1.91

Symbol definitions in the table: ρ represents explosive density, (kg/m³); D denotes detonation velocity, (m/s); A and B are coefficients of JWL equation of state (Pa); ω is dimensionless exponent; $R_{1}$ and $R_{2}$ are dimensionless parameters.

Numerical simulation results and validation

Figure 9 presents a comparison of the peak overpressure at various measurement points behind the wall, including the experimentally measured, numerically simulated, and CONWEP free-field theoretical values, to verify the reliability of the ALE multi-material numerical model employed in this study. The results indicate good overall agreement in the variation trend between the numerical simulation and the experimental measurements. The simulated peak overpressures are generally slightly lower than the measured values, with relative errors ranging from 27% to 40%, which falls within the commonly accepted range for numerical simulations in explosion mechanics.

Figure 9.

Comparative analysis of peak overpressure at gauges behind the wall.

Among these measurement points, T3 exhibits the smallest error (27%), and the error generally decreases with the increasing of scaled distance. After excluding the anomalous point T6 (affected by splashing fragments), the monotonic decay trend of the numerical simulation data is entirely consistent with the experimental data.Furthermore, the model accurately captures the local overpressure enhancement feature at point T4 caused by Mach reflection, validating its capability to simulate the coupling effects of shock wave diffraction and reflection.

Based on the validated numerical model, a quantitative analysis of the protective performance of the anti-blast wall was conducted. The results indicate that within the near-field region, $\bar{R}$ < 3 m/kg^1/3, the wall demonstrates a significant shock wave overpressure attenuation effect, with an average attenuation rate reaching 71.38%. In the far-field region, $\bar{R}$ > 4 m/kg^1/3, the average overpressure attenuation rate decreases to 40.44%. The average attenuation rate in the near-field region is approximately 1.76 times that of the far-field region. Overall, the overpressure attenuation efficacy of this anti-blast wall gradually decreases as the scaled distance increases. It exhibits more prominent protective advantages against close-in explosion threats and is suitable for emergency protection and temporary barrier projects around explosion hazard sources. It should be noted that the above quantitative results are obtained based on the 5.3 m long finite-width wall model used in this test. When adopting the simplified infinite-length wall model applied in the subsequent parametric analysis, the adverse superposition effect of lateral diffracted waves is eliminated, so the protective performance is expected to be further improved.

Due to the inherent numerical dissipation associated with the 3D mesh mapping technique of the ALE algorithm, this study did not perform a point-by-point quantitative comparison between the measured and simulated waveforms. The core objective of the numerical simulation was to capture the propagation timing and structural characteristics of the shock wave system. The dissipation of high-frequency components caused by mesh mapping primarily affects the amplitude and temporal details of the waveform but does not alter the physical mechanisms of wave system interaction. The model accurately reproduced the local enhancement effect caused by Mach reflection behind the wall and the four-peak diffraction structure associated with the variable cross-section wall, providing a reliable basis for subsequent mechanistic analysis.

Human injury assessment

Injury criteria

In explosion accidents, the trauma sustained by personnel can be categorized into two main types: direct injury and indirect injury. Direct injuries primarily include blast wave injury and fragment injury. The former is caused by structural tearing and stress damage within biological tissues resulting from the propagation of the explosion shock wave, while the latter arises from penetration of the human body by high-velocity projectiles or secondary fragments generated by the explosion. Indirect injuries encompass impact injuries resulting from the displacement and collision of the human body propelled by the shock wave, as well as other injuries caused by secondary explosion effects such as high temperatures and toxic gases.

In this experiment, a bare TNT charge was used, and the anti-blast wall is a non-fragmenting modular structure, effectively eliminating the risk of fragment injury. Indirect injuries, however, are influenced by the coupling of multiple factors such as explosive type, individual physiological parameters, and environmental variables, resulting in complex injury mechanisms and considerable uncertainty. Therefore, this study focuses on the direct injury effects of explosion shock waves for the assessment of protective efficacy.

It should be specifically noted that this study conducts personnel injury assessment based solely on peak overpressure, without considering the coupled injury effects of impulse. For explosion scenarios involving larger yields or closer standoff distances, the influence of impulse cannot be neglected. Future research could further investigate injury assessment based on the dual parameters of overpressure and impulse to improve protective design standards.

Current research indicates that the craniocerebral region is the primary target organ for fatalities caused by blast shock waves, and its injury mechanism exhibits significant organ specificity (Ganpule et al., 2013). Explosion shock waves can affect brain tissue through three pathways: thoracic wall transmission, direct cranial impact, and transmission via the ear canal. Among these, the transmission pathway via the ear canal can directly transfer shock wave energy into the intracranial space without penetrating the skull, leading to transient intracranial pressure changes and disruption of the blood-brain barrier. This is the primary mechanism responsible for causing mild to moderate brain injury (Phipps et al., 2020).

Based on this, the present study focuses on investigating the quantitative relationship between blast overpressure transmitted via the ear canal and the risk of brain injury. To accurately simulate the ear exposure conditions of a person in a standing posture, the height of all overpressure measurement points behind the wall in this study was uniformly set at 1.65 m (error range ±0.05 m). This height corresponds precisely to the average anatomical height of the ear for a standing adult male, allowing for an accurate representation of the head’s blast exposure level in a soldier’s combat posture.

This study employs the blast overpressure criterion for brain injury via ear canal transmission, established by Ward et al. (1980) (Table 5), to conduct a quantitative assessment of personnel injury risk. This criterion, developed based on extensive animal experiments and human anatomical data, has been widely applied in the field of protective facility efficacy evaluation. As shown in Table 5, when the peak blast overpressure exceeds 0.234 MPa, personnel are highly likely to sustain fatal injuries. When the peak overpressure falls within the range of 0.172∼0.234 MPa, personnel may potentially sustain fatal injuries. When the peak overpressure is below 0.172 MPa, personnel exhibit no significant injury. This criterion provides a unified quantitative standard for this study to assess the protective effect of the rapid-assembly anti-blast wall on personnel.

Table 5.

Injury criteria for intracranial pressure through ear canal transmission.

Peak Overpressure(MPa)	Injury severity	Injury condition
<0.172	Minor injury	Non-fatal
0.172 ∼ 0.234	Moderate injury	Potentially fatal
>0.234	Severe injury	Highly likely to be fatal

Safety zone delineation

Based on the aforementioned intracranial pressure injury criterion for transmission via the external auditory canal, this study employed numerical simulation methods to systematically calculate the distribution pattern of overpressure at different scaled distances behind the anti-blast wall and at a height of 1.65 m (corresponding to the average height of the human ear).

Figure 10 presents the variation curve of peak overpressure at different scaled distances behind the anti-blast wall with the width of 5.3 m under a 117.5 kg TNT explosion load. The figure shows that the peak overpressure behind the wall exhibits a significant monotonic decay trend with increasing scaled distance: as the scaled distance increases from 2.58 m/kg^1/3 to 4.83 m/kg^1/3, the peak overpressure decreases from 0.078 MPa to 0.043 MPa, resulting in an overall attenuation amplitude of 44.87%.

Figure 10.

Overpressure distribution behind the anti-blast wall.

A quantitative assessment was conducted by integrating the craniocerebral blast injury criteria from Table 5. The peak overpressure at all measurement points in this study, which ranged from 0.043 to 0.077 MPa, was significantly lower than the critical value for minor injury of 0.172 MPa, and fell far below the thresholds for moderate injury, defined as 0.172 to 0.234 MPa, and severe fatal injury, exceeding 0.234 MPa. This indicates that, under the explosion conditions set in this study, this type of anti-blast wall can effectively attenuate blast wave energy, controlling the risk of personnel craniocerebral injury via ear canal transmission to a minor injury level.

Based on the aforementioned research findings, a risk zoning map for personnel craniocerebral injury behind the wall was plotted (Figure 11). The figure visually demonstrates that, under the conditions of this study involving a 117.5 kg TNT explosion and a wall height of 4.42 m, the area beyond twice the wall height, i.e., approximately 8.84 m behind the wall, poses the minimum risk of blast wave hazard to personnel. This conclusion provides an important theoretical basis for personnel safety protection in explosion-related scenarios. It can be used to guide the rational planning of on-site safety zones, effectively reducing the risk of casualties caused by explosion accidents.

Figure 11.

Human injury distribution behind the wall.

Parametric study and predictive equation

To quantify the influence of wall geometric parameters and the scaled standoff distance between the wall and the detonation centre on the protective effectiveness, the following dimensionless parameters are defined:

(1) The scaled height $\bar{H}$ (m/kg^1/3):

\bar{H} = \frac{H}{\sqrt[3]{W}}

(3)

where H is the wall height, m; W is the TNT mass, kg.

(2) The scaled distance between the detonation centre and the wall $\bar{Z}$ (m/kg^1/3):

\bar{Z} = \frac{Z}{\sqrt[3]{W}}

(4)

where Z is the straight-line distance between the detonation centre and the wall, m.

(3) The scaled thickness of the upper section of the wall $\bar{T}$ (m/kg^1/3):

\bar{T} = \frac{T}{\sqrt[3]{W}}

(5)

where T is the thickness of the upper section of the wall, m.

The numerical analysis results indicate that the scaled height, scaled standoff distance, and scaled thickness are the core parameters influencing the protective performance of the anti-blast wall. In the subsequent analysis, the influence of each parameter on the protective effect will be analyzed through numerical simulation.

Scaled height

In this analysis, the wall thickness and the distance from the detonation centre to the front toe of the wall (5 m) in the numerical model are consistent with the prototype test. Considering that rapid-assembly anti-blast walls are typically deployed continuously along the perimeter of the protected area in practical engineering, their longitudinal length is much greater than their thickness, allowing the end-effects of shock wave diffraction to be neglected. Therefore, for this simulation, the wall is simplified as an infinite-length plane strain model, which better aligns with actual application scenarios. In the model, the thickness of the upper section of the wall is uniformly set to 2.12 m. Six groups of scaled wall height are defined: 0.45, 0.68, 0.90, 1.13, 1.35, and 1.58 m/kg^1/3. The layout of measurement points behind the wall is identical to the experimental setup, with all points at a height of 1.65 m, corresponding to the average height of the human head.

Figure 12 presents the comparison curves of peak overpressure at various measurement points behind the wall for different scaled wall heights. The figure shows that the scaled wall height is the dominant factor influencing the overpressure distribution behind the wall. The peak overpressure behind the wall exhibits a significant nonlinear decay trend as the scaled wall height increases. Under the most stringent near-field condition, corresponding to a measurement point scaled distance of 2.58 m/kg^1/3), when the scaled height $\bar{H}$ increases from 0.45 to 1.58, the peak overpressure behind the wall decreases from 39.38 kPa to 11.62 kPa, resulting in an overall attenuation amplitude of 70.49%. Using the free-field incident overpressure (161 kPa) calculated by the CONWEP code for the same condition as a baseline, the blast wave attenuation rate improves from 75.54% to 92.78%, representing a relative enhancement in blast wave mitigation efficacy by a factor of 1.23.

Figure 12.

Peak overpressure of varying scaled heights.

Further analysis reveals a threshold effect in the blast wave attenuation performance of the anti-blast wall. When the scaled height $\bar{H}$ increases from 0.45 to 0.90, the overpressure attenuation rate is the fastest, with the peak overpressure decreasing from 39.38 kPa to 18.57 kPa, an attenuation amplitude of 52.84%. However, when $\bar{H}$ continues to increase from 1.13 to 1.58, the overpressure attenuation rate slows down significantly, with the peak overpressure only decreasing from 13.94 kPa to 11.62 kPa, an attenuation amplitude of merely 16.64%. This indicates that when $\bar{H}$ exceeds 1.13, most of the incident shock wave has been effectively blocked by the wall, and further increasing the wall height yields a substantially reduced marginal benefit to the protective performance.

Under the condition of a fixed scaled wall height, the peak overpressure behind the wall decays slowly and monotonically as the scaled distance increases. The absolute overpressure attenuation is greater for the lower scaled height condition, reaching 4.46 kPa at a scaled height of 0.45, and smaller for the higher scaled height condition, reaching 2.62 kPa at a scaled height of 1.58. Overall, the influence of the scaled distance of the measurement point on the overpressure behind the wall is far less significant than that of the scaled wall height.

Scaled standoff distance

To further investigate the influence of explosion location on the protective performance of the rapid-assembly anti-blast wall, this analysis is based on the prototype test condition. The wall height (4.42 m) and upper section thickness (2.12 m) are kept constant. An infinite-length plane strain model is still employed. The layout of measurement points behind the wall is identical to that of the prototype test, with all points set at a height of 1.65 m, corresponding to the average human head height. A total of five scaled distances between the detonation centre and the wall are set: 0.82, 1.02, 1.23, 1.43, and 1.63 m/kg^1/3. The simulation results are shown in Figure 13.

Figure 13.

Peak overpressure of varying scaled standoff distances.

The results indicate that the scaled standoff distance not only determines the magnitude of the overpressure behind the wall but also significantly alters its spatial distribution pattern. When the scaled standoff distance $\bar{Z}$ ≤ 1.02 m/kg^1/3 (close-in explosion condition), the overpressure behind the wall shows a monotonic decay trend along the shock wave propagation direction, with the peak occurring at the nearest Gauge 1 behind the wall. However, when the scaled standoff distance $\bar{Z}$ ≥ 1.23 m/kg^1/3 (medium to far-field explosion condition), the distribution pattern of the overpressure behind the wall undergoes a fundamental change, exhibiting a unimodal characteristic of first increasing and then decreasing. The location of the peak shifts rearward from the near-wall Gauge 1 to the region of Gauges 3∼4. Taking the condition of $\bar{Z}$ = 1.43 m/kg^1/3 as an example, the peak overpressure at Gauge 3 reaches 18.10 kPa, which is 15.14% higher than that at the near-wall Gauge 1 (15.72 kPa). It then decreases to 17.07 kPa upon propagating to the farthest Gauge 7.

A quantitative marginal benefit analysis of the impact of scaled standoff distance on protective performance was conducted. When the scaled distance $\bar{Z}$ increased from 0.82 to 1.43, the maximum overpressure behind the wall only decreased from 18.86 kPa to 18.10 kPa, a reduction of 4.03%. However, when $\bar{Z}$ was further increased from 1.43 to 1.63, the maximum overpressure decreased by merely 0.18 kPa, a reduction of less than 1%. This indicates that when $\bar{Z}$ exceeds approximately 1.43, further increasing the scaled standoff distance yields an extremely limited improvement in protective performance.

Scaled thickness

To further investigate the influence of the scaled wall thickness on the distribution pattern of overpressure behind the wall, this analysis was also based on the prototype test condition. The wall height (4.42 m) and the distance from the detonation centre to the front toe of the wall (5 m) were kept constant. The same infinite-length plane strain model as described previously was employed. The layout of measurement points behind the wall was identical to the prototype test, with all points at a height of 1.65 m. Five groups of scaled wall thickness were set: 0.22, 0.33, 0.43, 0.54, and 0.65 m/kg^1/3. The simulation results are shown in Figure 14.

Figure 14.

Peak overpressure of varying scaled wall thicknesses.

The influence of the scaled wall thickness on the overpressure behind the wall exhibits a distinct segmental characteristic and a crossover feature. For the scaled thickness $\bar{T}$ ≤ 0.43 (thin-wall condition), the overpressure does not show a monotonic decay with increasing thickness. In the near-field region (measurement point scaled distance $\bar{R}$ ≤ 2.99 m/kg^1/3), the overpressure generally decreases as the thickness increases. Conversely, in the far-field region ( $\bar{R}$ > 2.99 m/kg^1/3), the overpressure slightly increases with greater thickness, resulting in the curves for different thicknesses intersecting at approximately $\bar{R}$ ≈ 2.99 m/kg^1/3. However, for $\bar{T}$ ≥ 0.43 (thick-wall regime), the crossover effect disappears. The overpressure shows a monotonic decay trend as the thickness increases, and the attenuation amplitude is greater in the near-field than in the far-field. At a scaled distance of $\bar{R}$ = 2.58 m/kg^1/3, increasing the scaled thickness from 0.43 to 0.65 reduces the overpressure from 18.57 kPa to 16.51 kPa, corresponding to an attenuation of 11.09%. In contrast, at $\bar{R}$ = 4.83 m/kg^1/3, the overpressure only decreases from 16.82 kPa to 15.94 kPa, representing a mere 5.23% attenuation.

A quantitative marginal benefit analysis of the protective performance was further conducted. When the scaled thickness $\bar{T}$ increased from 0.22 to 0.43, the maximum overpressure behind the wall decreased from 19.13 kPa to 18.57 kPa, representing a reduction of 2.93%. When $\bar{T}$ was further increased from 0.43 to 0.65, the maximum overpressure decreased to 16.56 kPa, corresponding to a reduction of 10.82%. However, it is important to clarify that, compared to the previously analyzed scaled wall height and scaled standoff distance between the detonation centre and the wall, the influence of the scaled thickness on the overpressure behind the wall is significantly smaller.

Mach Stem region characteristics

According to the cranial injury criterion of external auditory canal transmission listed in Table 5, the peak overpressure behind the blast wall varies from 9.00 kPa to 39.38 kPa under all test conditions. Such values are far below the critical threshold of 172 kPa that triggers minor cranial injury. Even under extreme working conditions involving the minimum wall height, the closest detonation distance and the thinnest wall with a scaled thickness of 0.22, the protective structure can effectively avoid cranial injury risks for personnel. Moreover, the relevant parameter exerts only a slight influence, which merely acts as a secondary factor governing blast protection performance.

This study investigates the evolution law and structural characteristics of shock wave systems behind variable cross-section blast walls. Although multi-path wave propagation in the shadow zone over unobstructed flat ground has been well established in atmospheric acoustics, as summarized by Attenborough (2014), relevant research on variable cross-section blast walls remains limited in explosion mechanics. For the first time, this work systematically observes and elaborates the unique four-peak diffraction mechanism triggered under hundreds-of-kilogram TNT explosion loads, and reveals the modulation effect of variable cross-section geometry on conventional wave system structures. Numerical simulations accurately reproduce the formation sequence and relative amplitude distribution of the four-peak waveform, offering profound understanding of shock wave diffraction characteristics of rapid-assembly variable cross-section blast walls.

A typical case with a scaled wall height of 1.35 m/kg^1/3 and a scaled distance between the detonation centre. The measuring point is located 12.64 m horizontally from the detonation center at a height of 1.65 m. This position corresponds to measuring point T3 in the test with a scaled distance of m/kg^1/3, which acts as a representative monitoring site in the near-field area. The overpressure-time history characteristics and the shock wave propagation and evolution process at a height of 1.65 m behind the wall were analyzed. The results are shown in Figures 15 and 16, respectively.

Figure 15.

Numerically simulated overpressure-time history curve.

Figure 16.

Time-sequenced pressure contour plots of shock wave propagation.

As shown in the overpressure-time history curve in Figure 15, the overpressure behind the wall under this condition exhibits a unique four-peak characteristic, which is significantly different from the Mach diffraction time history of conventional rectangular walls with a uniform cross-section (typically having only 2∼3 peaks). The initial segment of the curve corresponds to standard atmospheric pressure (101.3 kPa, absolute pressure). The first overpressure peak appears at 34 ms, followed by three secondary peaks at 42 ms, 49 ms, and 58 ms, respectively. The amplitude of each peak decays monotonically over time, and the entire shock wave loading process lasts approximately 31 ms.

Combined with the time-sequenced pressure contour plots of shock wave propagation in Figure 16, the formation mechanism and propagation process of the four-peak structure can be correlated as follows:

The first peak (t = 34 ms): Corresponds to the initial diffracted wave from the top of the wall. As seen in the pressure contour plots at 23.78 ms and 28.57 ms, the spherical shock wave generated by the explosion propagates outward. At approximately 28 ms, the shock wave has bypassed the wall and is about to travel toward the ground behind it, forming the first overpressure peak.

The second peak (t = 42 ms): Corresponds to the overpressure generated when the main diffracted wave from the wall top, after ground reflection, merges with the subsequent incident wave to form a Mach reflection, and passes the measurement point again. Its generation mechanism is illustrated in Figure 17. The pressure contour plot at 41.85 ms shows that the diffracted shock wave undergoes strong reflection upon reaching the ground, forming an upward-propagating reflected wavefront. This wavefront arrives at the measurement point at t ≈ 42 ms, forming the second overpressure peak.

Figure 17.

Schematic of Mach Stem formation.

The third peak (t = 49 ms): Corresponds to the secondary incident wave. This wave is formed when a secondary shock wave, generated by the ground reflection between the TNT charge and the blast-facing surface of the wall, diffracts over the top of the wall. This delayed secondary wavefront propagates to the measurement point behind the wall at approximately 49 ms, forming the third overpressure peak.

The fourth peak (t = 58 ms): Corresponds to the shock wave formed by the ground reflection of the secondary diffracted shock wave. As seen in the pressure contour plots at 52.52 ms and 59.79 ms, the secondary diffracted wave propagates to the ground and undergoes reflection, forming an upward-propagating reflected wavefront. This wavefront arrives at the measurement point at approximately 58 ms, forming the fourth overpressure peak.

Further analysis revealed that two distinct inflection points appear in the time-history curve at 36.3 ms and 42 ms. These are also attributed to the variable cross-section attribute of the wall. The first inflection point at 36.3 ms is caused by the local reflection of the main diffracted wave at the trapezoidal corner in the middle section of the wall, leading to a minor fluctuation in the overpressure at the measurement point. The second inflection point at 42 ms corresponds to the reflection effect of the main diffracted wave at the junction between the rear toe of the wall and the ground. Compared to a conventional wall with a uniform cross-section, the stepped structure of the variable cross-section wall decomposes the single top-diffracted wave into two independent wavefronts. This not only generates an additional overpressure peak but also introduces two characteristic inflection points.

Predictive formula for peak overpressure behind walls

To establish a quantitative predictive model for the peak overpressure at any measurement point behind a rapid-assembly anti-blast wall, the comprehensive scaled distance $\bar{Z_{0}}$ is defined as follows:

\bar{Z_{0}} = \frac{d_{1} + d_{2} + T}{\sqrt[3]{W}}

(6)

where d₁ is the horizontal standoff distance from the detonation centre to the blast-facing surface of the anti-blast wall, m; d₂ is the horizontal distance from the measurement point to the back face of the anti-blast wall, m; T is the thickness of the anti-blast wall, m; W is the TNT mass, kg.

Based on the similarity law theory of blast wave diffraction over rigid barriers and existing experimental studies (Hajek and Foglar, 2015), the peak overpressure P_s behind the wall is governed by three core parameters: the comprehensive scaled distance $\bar{Z_{0}}$ , the wall height H₁, and the measurement point height H₂. That is:

P_{s} = f (\bar{Z_{0}}, H_{1}, H_{2})

(7)

To more accurately characterize the protective performance of the anti-blast wall and to simplify the overpressure prediction model, the comprehensive scaled distance is modified using the wall height H₁ and the measurement point height H₂. This leads to the definition of the protection efficacy coefficient K, as follows:

K = \frac{\bar{Z_{0}} \times H_{1}}{H_{2}}

(8)

To further eliminate the coupling effects of geometric parameters and facilitate obtaining a unified-form overpressure prediction formula through nonlinear regression fitting, the modified peak overpressure $P_{s}^{'}$ is defined as follows:

P_{s}^{'} = \frac{P_{s} \times H_{2}}{(d_{1} + d_{2} + T) \times H_{1}}

(9)

The previous analysis of the shock wave diffraction mechanism indicates that the wall height is the dominant factor influencing the overpressure distribution behind the wall. Since the overpressure values in the far-field region are extremely low and typically do not pose a personnel injury risk, this study considers only the medium to near-field region where the comprehensive scaled distance $\bar{Z_{0}}$ ≤ 4 m/kg^1/3. After substituting the simulation data within this range into equations (8) and (9) for parameter conversion, a power function model was employed for nonlinear regression fitting, yielding the optimal fitting curve:

P_{s}^{'} = 0.022843 K^{- 1.854033}

(10)

Based on the nonlinear regression results of the power function model, the coefficient of determination (R²) for this predictive formula is 0.95, indicating an excellent fitting performance. The fitting results of the formula for the numerical simulation conditions are shown in Figure 18. It should be noted that the applicable scope of this predictive formula is limited to: the variable cross-section rapid-assembly anti-blast wall with a trapezoidal lower section and rectangular upper section studied in this paper. Furthermore, it is only applicable to medium to near-field conditions where the comprehensive scaled distance $\bar{Z_{0}}$ ≤ 4 m/kg^1/3.

Figure 18.

Simulated values and fitting curve.

Finally, it should be noted that equations (6)–(9) belong to an empirical regression model based on experimental data, not a scaling-law model in the strict sense. This leads to non-physical divergence behavior (e.g., $P_{s}^{'}$ →∞) under physical limits (e.g., H₁→0 or H₂→0), and the regression coefficients contain implicit units, limiting arbitrary extrapolation beyond the calibration dataset. Furthermore, a mathematical coupling exists in the model: the geometric term (d₁+d₂+T) appears in both the independent and dependent variables. While this construction improves the goodness-of-fit to the current experimental data, it reduces the statistical independence of the variables. For limit conditions or extreme scaling, validated numerical simulation methods should be prioritized.

Conclusions

This study conducted field experiments, numerical simulations, and theoretical calculations on rapid-assembly anti-blast walls under the scenario of a 117.5 kg TNT near-ground explosion. It systematically investigated the blast wave attenuation performance, influencing parameters, and prediction methods. The main conclusions are as follows:

(1) The rapid-assembly anti-blast wall exhibits excellent blast wave attenuation performance. Under a 117.5 kg TNT explosion load, the wall did not overturn or undergo global displacement, with only localized charring observed on the blast-facing surface. The overpressure behind the wall shows a nonlinear decay with increasing distance, achieving a maximum attenuation rate of 78.2%.

(2) A numerical model was established, and the spatial variation of the protective efficacy was quantified. The LS-DYNA based numerical model could satisfactorily replicate the experimental results, with a simulation error for the peak overpressure ranging from 27% to 40%, meeting the accuracy requirements. The average overpressure attenuation rate reached 71.38% in the near-field region ( $\bar{R}$ < 3 m/kg^1/3) and decreased to 40.44% in the far-field region ( $\bar{R}$ > 4 m/kg^1/3). The near-field protective efficacy is approximately 1.76 times that of the far-field.

(3) The influence weight and threshold effects of various parameters on the protective efficacy were clarified. The scaled wall height is the dominant factor. The overpressure shows a significant nonlinear decay as it increases, exhibiting a threshold: the attenuation is fastest within the range of 0.45 ∼ 1.13 m/kg^1/3, beyond which the marginal benefit plummets. The scaled distance between the detonation centre and the wall is the secondary factor; the benefit of increasing the distance becomes negligible beyond 1.43 m/kg^1/3. The scaled wall thickness is a minor factor, with a near-far field crossover effect observed in the thin-wall regime.

(4) The distinctive four-peak diffraction modulation mechanism of variable cross-section blast walls is revealed. Unlike uniform cross-section walls that generally show two to three peaks, the overpressure time history behind the variable cross-section wall displays a characteristic four-peak structure at the near-field observation height of 1.65 m. The four peaks correspond successively to the initial top diffracted wave, Mach wave generated by ground reflection of the main diffracted wave, secondary reflected wave on the blast-facing surface, and ground reflected wave of the secondary diffracted wave. The stepped geometric structure does not change the fundamental formation mechanism of wave systems, yet it introduces additional reflection interfaces to effectively adjust the amplitude and phase difference of each subwave, which eventually gives rise to the multi-peak structure.

(5) A quantitative predictive formula for the peak overpressure behind the variable cross-section rapid-assembly anti-blast wall was established. This formula is derived by fitting a power function relationship between the protection efficacy coefficient and the modified peak overpressure behind the wall, achieving a coefficient of determination R² = 0.95. It is applicable to medium and near-field conditions where the comprehensive scaled distance $\bar{Z_{0}}$ ≤ 4 m/kg^1/3. This formula enables rapid estimation of the peak overpressure behind the wall, providing a convenient quantitative calculation method for evaluating the protective efficacy of this type of anti-blast wall.

This study considered only the injury effects of peak overpressure and did not address the coupled effects of impulse. Future work will involve injury assessment based on the dual parameters of overpressure and impulse.

Footnotes

ORCID iDs

Guoan Liu

Peng Wang

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research is funded by the State Key Laboratory of Disaster Prevention & Mitigation of Explosion & Impact (NO. 20260106).

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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